Linear Extensions of Isometries between Groups of Invertible Elements in Banach Algebras

We show that if T is an isometry (as metric spaces) from an open subgroup of the invertible group A of a unital Banach algebra A onto an open subgroup of the invertible group B of a unital Banach algebra B, then T is extended to a real-linear isometry up to translation between these Banach algebras. We consider multiplicativity or unti-multiplicativity of the isometry. Note that a unital linear isometry between unital semisimple commutative Banach algebra need be multiplicative. On the other hand, we show that if A is commutative and A or B are semisimple, then (T (eA)) T is extended to a isometrical real algebra isomorphism from A onto B. In particular, A is isometric as a metric space to B if and only if they are isometrically isomorphic to each other as metrizable groups if and only if A is isometrically isomorphic to B as a real Banach algebra; it is compared by the example of Żelazko concerning on non-isomorphic Banach algebras with homeomorphically isomorphic invertible groups. Maps between standard operator algebras are also investigated.